A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Math clubs and their potentials: making mathematics fun and exciting. A case study of a math club

The purpose of this study is to describe the results obtained from a survey whose goal was to examine the combination of variables that have contributed to the success of a middle school math club. This is a case of a middle school in which the students are extremely successful in mathematics, and where the majority of the students voluntarily attend its math club. The results of the study show that the students have positive attitudes about mathematics and the club, and that some of the reasons that influenced them to attend the club were those of being with friends and eating donuts at the club. The results were similar for students of both genders and all grades. In addition, since positive attitudes are associated with higher levels of math achievement, such clubs have the potential to encourage students to enrol in additional mathematics classes while in high school, as well as pursuing mathematics related careers.     

From formal standards to everyday practice of mathematics learning

This paper uses the example of six Japanese teachers and their mathematics lessons to illustrate how clear, high standards for mathematics instruction are combined with teachers' holistic concern for students. We draw upon data from the Third International Math and Science Study Case Study Project in Japan that was designed to elucidate the context behind the high achievement of Japanese students. Using everyday examples of classroom practice, we illustrate both flexibility in teachers' approach to teaching and adherence to Monbusho's (Ministry of Education, Science, Sports, and Culture)Course of Study. Our purpose is to emphasize how flexibility and attention to individual needs by Japanese teachers combine with quality mathematics instruction based on the detailed Japanese curricula. Six teachers' characteristics and lessons (two teachers at each educational level—elementary, junior high, and high school) are described in order to show the variety of teachers who exist in Japan. These teachers use their understanding of theCourse of Study and are supported by their school environment to enhance their students' conceptual understanding of the fundamentals of mathematics. Characteristics of their teaching include: 1) involving the whole class in learning. 2) using extremely focused curriculum guidelines that expect mastery of concepts at each grade level, 3) thoroughly covering mathematics units in an organized and in-depth manner, 4) leading classes as facilitators or guides more often than as lecturers, and 5) focusing on problem solving with the primary goal of developing students' ability to reason, especially to reason inductively. The examples in this paper show how these methods develop in individal classrooms.     

Incentivizing advanced mathematics study at upper secondary level: the case of bonus points in Ireland

Secondary level mathematics education in Ireland has recently experienced a period of significant change with the introduction of new curricula and the addition of an incentive to study upper secondary mathematics at the most advanced level (Higher Level). This incentive, typically referred to as ‘bonus points’, appears to have aided a significant increase in the number of students studying upper secondary mathematics at Higher Level. However, thematic analysis of interviews with experienced upper secondary mathematics examiners and exploration of mathematics diagnostic test data outlined in this paper suggest that the difficulty of the Higher Level upper secondary mathematics final examination in Ireland has reduced since the introduction of the bonus points initiative. The sharp increase in students attempting this examination coupled with a policy of maintaining a consistent proportion of students achieving passing grades was identified as a key reason for this possible reduction in standards.     

The development of reasoning skills during compulsory 16 to 18 mathematics education

The belief that studying mathematics improves reasoning skills, known as the Theory of Formal Discipline (TFD), has been held since the time of Plato. Research evidence supports this idea, at least in the context of students who had chosen to study post-compulsory mathematics. Here we examined the development of reasoning skills in 16- to 18-year-old Cypriot students, who are required to study mathematics until age 18. One hundred and eighty-eight students, studying high- or low-intensity mathematics, completed the abstract Conditional Inference Task and the contextual Belief Bias Syllogisms task at ages 16, 17 and 18. While the high-intensity group improved on the conditional inference task and showed a reduction in belief bias, the low-intensity group did not change on either measure. This is promising for the TFD, but suggests that a certain level of mathematical study may be necessary for students' general reasoning skills to develop.     

From lessons to lectures: NCEA mathematics results and first-year mathematics performance

Given the recent radical overhaul of secondary school qualifications in New Zealand, similar in style to those in the UK, there has been a distinct change in the tertiary entrant profile. In order to gain insight into this new situation that university institutions are faced with, we investigate some of the ways in which these recent changes have impacted upon tertiary level mathematics in New Zealand. To this end, we analyse the relationship between the final secondary school qualifications in Mathematics with calculus of incoming students and their results in the core first-year mathematics papers at Canterbury since 2005, when students entered the University of Canterbury with these new reformed school qualifications for the first time. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.     

Stepping beyond high school mathematics: a case study of high school women

The Summer Explorations and Research Collaborations for High School Girls (SEARCH) Program, held annually since 2004 at Mount Holyoke College in the US, was created for talented high school girls to explore mathematics beyond that taught in high school. Our study, which focuses on factors that facilitate or inhibit the pursuit of higher level mathematics by girls, is centered on the 2006 SEARCH Program. We present a combination of qualitative and quantitative data drawn from student journals written during SEARCH, program evaluations written at the end of SEARCH, post-program interviews, and comparisons with two peer group samples. From this data we point to important factors, such as developing a mathematical voice, gaining a broader view of advanced mathematics, being challenged in a supportive atmosphere, and having a positive stance toward risk-taking, that may help to maintain the interest of talented girls in advanced mathematical studies.     

The Brazilian mathematics textbook assessments

This paper briefly reviews Brazilian textbook policies during the twentieth century, and pays careful attention to its latest development, PNLD—Programa Nacional do Livro Didático (National Textbook Program) and its textbook assessments, which selects textbooks that are freely distributed by the Ministry of Education (MEC) to students of public elementary, middle and high schools. We focus on the mathematics textbook assessments, describing their evolution and commenting on some of their accomplishments and problems. The first assessment was carried out for the 1997 school year and retained basically its initial formulation, with changes and improvements, til the assessment for the 2018 school year. As of now, the program is undergoing substantial changes which are briefly discussed and which worry many educators. This assessment was responsible for a definite improvement of mathematics textbooks for elementary, middle and high schools in Brazil. It constitutes a good example of a successful program, coordinated by competent persons from mathematics education, schools of education and mathematics departments of good universities, at first without political interference, and strongly backed by powerful officials in MEC. In the last few years, pressures from publishers and conservative groups have undermined the assessments, with possibly serious consequences.     

The influence of overcoming fixation in mathematics towards divergent thinking in open-ended mathematics problems on Japanese junior high school students

The purpose of this study is to find the influence of overcoming fixation in mathematical problem-solving towards divergent thinking in open-ended mathematics problems. This paper presents findings of investigation of Japanese junior high school students. Students who can overcome fixation in mathematics can contribute varied and original ideas in open-ended situations in mathematics.     

Measuring early mathematics knowledge via number skills and task types

Early number skills are a critical aspect of early mathematics development. However, the constructs that comprise early number skills differ across assessments, and previous studies have proposed various models of early mathematics skills comprised of formal and informal tasks. This study explored the factor structure of a researcher-developed measure of mathematics administered to a large, geographically diverse sample of kindergarten students at risk for mathematics difficulty (n = 580) in a randomized control trial. Consistent with previous research, factors representing early number skills and task types emerged. Importantly though, the best fitting model was one in which both skill types (e.g., number identification, magnitude comparison) and task types (i.e., informal and formal) were modeled. The inclusion of task type as a factor in early mathematics assessment has many potentially important ramifications. Recommendations for attending to task types when assessing early number skills, and implications for instruction and measurement are discussed.     

Case study projects for college mathematics courses based on a particular function of two variables

Based on a sequence of number pairs, a recent paper (Mauch, E. and Shi, Y., 2005, Using a sequence of number pairs as an example in teaching mathematics, Mathematics and Computer Education, 39(3), 198–205) presented some interesting examples that can be used in teaching high school and college mathematics classes such as algebra, geometry, calculus, and linear algebra. In this paper, this study is generalized further to develop a few interesting case study proposals that can be used for student projects in college mathematics courses such as real functions, analytic geometry, and complex variables. In addition to using them in individual courses, these studies may also be combined to offer seminars or workshops to college mathematics students. Projects like these are likely to promote student interest and get students more involved in the learning process, and therefore make the learning process more effective.     

Reorganizing freshman business mathematics II: authentic assessment in mathematics through professional memos

The first part of this two-part paper (13) described the developmentof a new freshman business mathematics (FBM) course at our college.In this paper, we discuss our assessment tool, the businessmemo, as a venue for students to apply mathematical skills,via mathematical modelling, to realistic business problems.These memos have proven a crucial step in turning our FBM coursearound from a dreaded course with little connection to students’intended careers into a course where students experience thepower of mathematics for solving problems and informing decisions.Comments from students in the course throughout its 6-year historyclearly point to the course's value and importance.     

A Sino-German semi-virtual seminar in mathematics education

In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.     

The concept of invariance in school mathematics

In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change leaves some relationships or properties invariant, these properties prove to be inherently interesting to teachers and students.     

An ‘integrating’ application of mathematics

This paper presents a coherent application of several mathematical tools and concepts in an area of financial analysis. As an educational tool, the example demonstrates the usefulness of seemingly obscure mathematics to non‐mathematical students, and provides an interesting case study or exercise to more advanced students of mathematics.     

A hybrid model of mathematics support for science students emphasizing basic skills and discipline relevance

The problem of students entering university lacking basic mathematical skills is a critical issue in the Australian higher-education sector and relevant globally. The Maths Skills programme at La Trobe University has been developed to address under preparation in the first-year science cohort in the absence of an institutional mathematics support centre. The programme was delivered through first-year science and statistics subjects with large enrolments and focused on basic mathematical skills relevant to each science discipline. The programme offered a new approach to the traditional mathematical support centre or class. It was designed through close collaboration between science subject coordinators and the project leader, a mathematician, and includes resources relevant to science and mathematics questions written in context. Evaluation of the programme showed it improved the confidence of the participating students who found it helpful and relevant. The programme was delivered through three learning modes to allow students to select activities most suitable for them, which was appreciated by students. Mathematics skills appeared to increase following completion of the programme and student participation in the programme correlated positively and highly with academic grades in their relevant science subjects. This programme offers an alternative model for mathematics support tailored to science disciplines.     

Mexican high school students' social representations of mathematics,its teaching and learning

This paper reports a qualitative research that identifies Mexican high school students’ social representations of mathematics. For this purpose, the social representations of ‘mathematics’, ‘learning mathematics’ and ‘teaching mathematics’ were identified in a group of 50 students. Focus group interviews were carried out in order to obtain the data. The constant comparative style was the strategy used for the data analysis because it allowed the categories to emerge from the data. The students’ social representations are: (A) Mathematics is…(1) important for daily life, (2) important for careers and for life, (3) important because it is in everything that surrounds us, (4) a way to solve problems of daily life, (5) calculations and operations with numbers, (6) complex and difficult, (7) exact and (6) a subject that develops thinking skills; (B) To learn mathematics is…(1) to possess knowledge to solve problems, (2) to be able to solve everyday problems, (3) to be able to make calculations and operations, and (4) to think logically to be able to solve problems; and (C) To teach mathematics is…(1) to transmit knowledge, (2) to know to share it, (3) to transmit the reasoning ability, and (4) to show how to solve problems.     

Images of mathematicians: a new perspective on the shortage of women in mathematical careers

Though women earn nearly half of the mathematics baccalaureate degrees in the United States, they make up a much smaller percentage of those pursuing advanced degrees in mathematics and those entering mathematics-related careers. Through semi-structured interviews, this study took a qualitative look at the beliefs held by five undergraduate women mathematics students about themselves and about mathematicians. The findings of this study suggest that these women held stereotypical beliefs about mathematicians, describing them to be exceptionally intelligent, obsessed with mathematics, and socially inept. Furthermore, each of these women held the firm belief that they do not exhibit at least one of these traits, the first one being unattainable and the latter two being undesirable. The results of this study suggest that although many women are earning undergraduate degrees in mathematics, their beliefs about mathematicians may be preventing them from identifying as one and choosing to pursue mathematical careers.     

The impact of metacognitive practice and teacher feedback on academic achievement in mathematics

The purpose of this study is to determine the effects of metacognitive strategies and content‐specific feedback on student achievement in high school mathematics. Participants consisted of a convenience sample of students in honors geometry. An analysis of variance with repeated measures was employed to address the research questions. For both the posttest and retention test, students in the experimental group significantly outperformed the students in the comparison group. However, students scored lower in the retention test than the posttest. The findings of this study offer a modest contribution to the body of empirical research on the impact of metacognitive practice and content‐specific feedback on academic achievement at the high school level.     

The history of mathematics as a coupling link between secondary and university teaching

During the years they spend in university, many mathematics students develop a very poor conception of mathematics and its teaching. This fact is bad in all cases, but even more in the case of those students who will be mathematics teachers in school. In this paper it is argued that the history of mathematics may be an efficient element to provide students with flexibility, open-mindedness and motivation towards mathematics. The theoretical background of this work relies both on recent research in mathematics education and on papers written by mathematicians of the past. Opinions are supported with examples. One example concerns a historical presentation of ‘definition’; it was developed with mathematics students who will become mathematics teachers. For students oriented to research or to applied mathematics, an example is presented to address the problem of the secondary-tertiary transition.